Binomial theorem problem on the terms of an expansion

In summary, the conversation is about finding an approximation of (0.99)5 using the first three terms of its expansion. The solution involves using the binomial theorem and calculating the three terms T1, T2, and T3. Adding these terms may not give the exact answer, but it satisfies the requirements of the problem. There is a question about whether using (0.9+0.09) would give the same answer, and it is stated that it would not necessarily give the same answer.
  • #1
agnibho
46
0

Homework Statement


Find an approximation of (0.99)5 using the first three terms of its expansion.

2. The attempt at a solution
To get to the binomial theorem I divided 0.99 into
(0.99)5 = (1-0.01)5 = {1+(-0.01)}5
Then,
T1 = 5C0(1)5 = 1 x 1=1
T2 = 5C1(1)5-1(-0.01)1 = 5x1x -0.01= (-0.05)
T3 = 5C2(1)5-2(-0.01)2 = 10 x 1x 0.0001 = (0.001)

Now do I add them?? But adding them doesn't get me to the answer. Please help. Thanks in advance.
 
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  • #2
agnibho said:

Homework Statement


Find an approximation of (0.99)5 using the first three terms of its expansion.

2. The attempt at a solution
To get to the binomial theorem I divided 0.99 into
(0.99)5 = (1-0.01)5 = {1+(-0.01)}5
Then,
T1 = 5C0(1)5 = 1 x 1=1
T2 = 5C1(1)5-1(-0.01)1 = 5x1x -0.01= (-0.05)
T3 = 5C2(1)5-2(-0.01)2 = 10 x 1x 0.0001 = (0.001)

Now do I add them?? But adding them doesn't get me to the answer. Please help. Thanks in advance.

Why do you say "adding them doesn't get me to the answer"? Of course, it does not get you to the exact value of 0.99^5, but that is not the issue. Just adding the terms you have fulfills all the requirements of the problem.

RGV
 
  • #3
OK thanks for the help! Actually I had some confusion about that operation.
Um...I's thinking that will I get to the same answer if I had divided 0.99 into (0.9+0.09)?
 
  • #4
You would get another approximation, not necessarily the same answer. You would have to calculate the fifth, fourth, and third powers of .9 which is harder than the same powers of 1!
 
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Related to Binomial theorem problem on the terms of an expansion

1. What is the Binomial Theorem?

The Binomial Theorem is a mathematical formula that allows us to expand a binomial expression raised to a power. It states that (a + b)^n = ∑(k=0 to n) nCk * a^(n-k) * b^k, where a and b are constants, n is the power, and nCk (read as "n choose k") is the binomial coefficient.

2. How do you solve a Binomial Theorem problem on the terms of an expansion?

To solve a Binomial Theorem problem, you first need to identify the given values for a, b, and n. Then, use the formula (a + b)^n = ∑(k=0 to n) nCk * a^(n-k) * b^k to expand the binomial expression. Simplify the terms and combine like terms to get the final expanded form.

3. What is a binomial coefficient?

A binomial coefficient is a number that represents the number of ways to choose a subset of k elements from a set of n elements. It is represented by nCk and is calculated by the formula nCk = n! / (k! * (n-k)!), where n! represents n factorial (n * (n-1) * (n-2) * ... * 2 * 1).

4. Can the Binomial Theorem be used for non-integer powers?

Yes, the Binomial Theorem can be extended to non-integer powers using the Binomial Series. The Binomial Series states that (a + b)^x = ∑(k=0 to ∞) (xCk * a^(x-k) * b^k) / k!, where x is any real number and xCk is the generalized binomial coefficient.

5. What are some real-life applications of the Binomial Theorem?

The Binomial Theorem has various applications in fields such as physics, engineering, and finance. It is used to calculate probabilities in statistics, to expand polynomials in algebra, and to approximate values in calculus. It also has applications in computer science and cryptography, where it is used in algorithms for data compression and encryption.

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